Variational principles for self-adjoint operator functions arising from second order systems

Birgit Jacob, Matthias Langer, Carsten Trunk

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Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form ⟨z''(t),y⟩ + d[z'(t), y] +a0[z(t), y] = 0. Here a0 and d are densely defined, symmetric and positive sesquilinear forms on a Hilbert space H. We associate with the variational evolution equation an equivalent Cauchy problem corresponding to a block operator matrix A , the forms t(λ)[x, y] := λ2⟨x, y⟩ + λd[x, y] +a0[x, y], where λ ∈ ℂ and x, y are in the domain of the form a0, and a corresponding operator family T(λ). Using form methods we define a generalized Rayleigh functional and characterize the eigenvalues above the essential spectrum of A by a min-max and a max-min variational principle. The obtained results are illustrated with a damped beam equation.
Original languageEnglish
Pages (from-to)501-531
Number of pages31
JournalOperators and Matrices
Issue number3
Publication statusPublished - 30 Sept 2016


  • block operator matrices
  • variational principle
  • operator function
  • second-order equation
  • spectrum
  • essential spectrum
  • sectorial form


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